Bondi mass
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| Bondi mass | |
|---|---|
| Name | Bondi mass |
| Units | joule per c^2 (mass-energy) |
| Symbols | M_B |
| Introduced | 1962 |
| Introduced by | Hermann Bondi, Rainer K. Sachs, Julian Schwinger |
Bondi mass The Bondi mass is a measure of the total mass–energy content of an isolated gravitating system as seen by observers at future null infinity. It was introduced in the early 1960s in studies of gravitational radiation and is central to the understanding of energy loss by outgoing gravitational waves in the context of general relativity. Developed within the work of Hermann Bondi and Rainer K. Sachs, the concept links gravitational radiation, asymptotic symmetries, and global conservation laws.
Definition and physical interpretation
The Bondi mass quantifies the remaining mass of an isolated system after accounting for energy carried away by gravitational radiation, electromagnetic radiation, and massless fields. In the setting of an isolated radiating source such as those considered in the analyses of Hermann Bondi and Rainer K. Sachs, the Bondi mass is defined at cuts of future null infinity and decreases monotonically along retarded time due to emitted radiation. Physically it is used in studies of binary inspirals analyzed by the LIGO Scientific Collaboration, comparisons with post-Newtonian results used by the European Gravitational Observatory, and in theoretical work by Roger Penrose and Stephen Hawking on global properties of space‑time.
Mathematical formulation
Mathematically, the Bondi mass is given by a surface integral over a sphere at future null infinity involving metric coefficients in an asymptotic expansion of the spacetime metric in Bondi–Sachs coordinates. The integrand contains components of the Bondi–Sachs mass aspect and the shear of outgoing null hypersurfaces, quantities that feature in the work of Rainer K. Sachs and Ezra T. Newman. Calculations often exploit tetrad formalisms developed by Ted Newman and Roger Penrose, and use techniques related to the Newman–Penrose scalars and the peeling theorem formulated in the literature of Andrzej Trautman and James W. York. The Bondi mass M_B(u) at retarded time u is typically expressed in terms of the mass aspect m(u, θ, φ) integrated over a unit sphere, with the news function N(u, θ, φ) controlling the flux.
Asymptotic flatness and Bondi–Sachs framework
The Bondi mass presupposes an asymptotically flat spacetime structure as formalized in the Bondi–Sachs framework, which was elaborated by Hermann Bondi, Rainer K. Sachs, and others including Ezra T. Newman. The framework selects outgoing null coordinates (u, r, θ, φ) and imposes boundary conditions at future null infinity I^+, a construction used in the conformal compactification approach pioneered by Roger Penrose. Asymptotic symmetries at I^+ form the Bondi–Metzner–Sachs group, discovered by Bondi, van der Burg, Metzner, and Sachs, and later explored by Andrew Strominger in connections with soft theorems and memory effects. The choice of cuts of I^+ and the BMS supertranslation freedom affects the mass aspect and requires care when comparing different observers such as those considered by Subrahmanyan Chandrasekhar and Kip Thorne.
Conservation laws and mass loss formula
The Bondi mass obeys a mass loss formula that equates the retarded-time derivative of the Bondi mass to a negative definite flux integral of the squared news function and contributions from outgoing matter fields. This result, central to the analysis of gravitational radiation by Bondi and Sachs, provides the rigorous statement that gravitational waves carry positive energy as argued by Richard Feynman in early heuristic discussions and later formalized within proofs by Edvard M. Lifshitz and Lev Landau style treatments. The positivity of Bondi mass under suitable conditions was proven in extensions of the positive energy theorem by Richard Schoen and Shing-Tung Yau and by Edward Witten in approaches related to spinors; these proofs relate to the manifold techniques used by Michael T. Anderson and Robert Bartnik.
Relation to ADM mass and Komar mass
The Bondi mass is distinct from the ADM mass defined at spatial infinity by Arnowitt, Deser, and Misner, and from the Komar mass defined for stationary spacetimes by Arthur Komar. For asymptotically flat, nonradiating stationary solutions such as the Schwarzschild solution studied by Karl Schwarzschild or the Kerr solution by Roy Kerr, all three notions coincide when evaluated appropriately, connecting work by Werner Israel on uniqueness theorems and by Brandon Carter. In dynamic, radiating situations the Bondi mass is generally less than or equal to the ADM mass; the difference equals the total energy radiated to future null infinity as computed in analyses by LIGO, Virgo, and pulsar timing collaborations.
Applications and examples
Bondi mass is applied in modeling gravitational-wave emission from compact binaries examined by the LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA, in analytic post-Newtonian approximations used by Thibault Damour, in numerical relativity simulations by the Simulating eXtreme Spacetimes (SXS) Collaboration, and in perturbative studies around black hole solutions explored by Subrahmanyan Chandrasekhar. Example spacetimes with explicit Bondi mass computations include linearized radiating solutions, Robinson–Trautman spacetimes named for Ivor Robinson and Andrzej Trautman, and isolated horizon treatments by Abhay Ashtekar. Observational implications tie to energy flux estimates relevant to the Nobel Prize–winning detections by the LIGO team.
Generalizations and extensions
Generalizations of Bondi mass address asymptotic structures in higher dimensions studied by Gary T. Horowitz, AdS/CFT contexts influenced by Juan Maldacena, and formulations including matter fields, electromagnetic contributions treated since the work of Hermann Weyl and James Clerk Maxwell, and quantum corrections considered in semiclassical gravity literature involving Stephen Hawking. Extensions also include bonding mass definitions under modified gravity theories explored by Cliff Will, and refined treatments of asymptotic symmetries and soft theorems pursued by Andrew Strominger, Monica Guica, and related researchers. Ongoing work connects Bondi mass with holographic energy constructs, memory effects investigated by Alessandra Buonanno, and conservation laws in gauge/gravity dualities examined by Edward Witten and Juan Maldacena.